Kid won't scream and cry every night, he's just seeing if that works to get his PS.  It's a tantrum.  Dad is backing you. Though to be fair it should be Dad doing the parenting but he's too lazy.

You shouldn't be doing the parenting it's not your job, you are Dad's gf but he's happy to do no real parenting and let you be the bad guy and he is the good guy.  I'd be careful because he is putting you in the role of evil SM.  He's already saying "It's your rule"

Won't be long before he is going behind your back and undermining you by letting kid do what he wants.

You will end up resentful.

Don't care if the kid is up all night gaming, his dad doesn't care.  Don't care if he is a picky eater - his Dad doesn't care.

I'd advise you to go back to being the gf of Dad and not the unappreciated workhorse and unpaid nanny.  Move back out if you can and date.

You can't care more than the REAL parent, who is in the home giving the kid the very clear impression that he doesn't agree with your rules.

I'm not sure why you want a future with a guy who doesn't give a rat's a$$ if his son is doing his schoolwork or gaming all night at age 10, but that's your prerogative.  If you do stay, I'd get used to having an unparented kid in your home, because your BF is too lazy and uncaring to make sure his has routines and structure.

You are being used by your BF.

Time to disengage from both your BF and his spoiled son.

Repeat as needed: Not my circus, not my monkey.

Move out until dad learns to parent his child.  With you there, he can avoid his responsibility to his child to be a parent and then have a built in scapegoat (hint: YOU) as to why his son turns out to be a feral person.

In the meantime, as you see it's you who is left feeling bad and can't sleep, not fear old dad.  
 

Also read 'StepMonster' by Wednesday Martin to see where your BFs lack of parenting and poor treatment of you will lead.

I think that it is easy for men to become Disney dad's. I see a lot of dad's have 2 modes: Fun dad and mean dad. There is nothing in between. 

You are new to the scenario and are creating a "normal" home, structure, routine, etc- all the things that women do more naturally. 

I think it is great that he is backing you up, that is a good sign. SS is pushing his boundaries- that is to be expected. Things will settle once he gets the hang of how things are going to be now. The less you put up with in regards to the tantrum, the more you ignore it and just have a don't care attitude about his tantrum, the better off you will be. All parents will have things that they have to change because things aren't working like they should from time to time. Kids will rebel if it makes their life less fun, part of being a parent is realizing that we aren't doing our jobs if kids lives are just fun. They need to learn good sleeping habits, responsibility, study habits, etc. 

BF would also see it as me failing the mom test. 

Sorry, what? You are not the mom.

I know exactly how you feel. At this point you feel like you have no idea to find a happy medium and or what will work in your home that would make everyone happy, along with your BF to back you up (without looking like the bad guy), right?

But at this time, he already sees how you set rules and trying to instill structure for a young boy. It's not easy, but it already looks like you are on the toad to being painted as the bad guy, since dear ole dad is the "friend" and not the parent! He is too scared to parent his own child without thinking he will be looked at as the bad guy for setting rules too! So, of course, he has you to look like that and he can still remain the "friend" in all this. 
 

I know exactly how you feel, because my SS is 9 going on 10 and he is: picky, face to a screen ALL THE TIME, doesn't know simple things like (wash up before and after meals, pick up your plate and any mess around it, turn off lights, lift the toilet seat, pick up your mess (clothes off the floor) and is also a "picky eater" (eats junk, junk, sugar, sugar and JUNK). I am looked at as the mean SM who has rules and makes SS do all these awful things. According to his father I "order him around" because I ask those simple things a 9/10 year old boy to do!!! I never ask his son to do chores or clean my home and or do anything that would "harm" his sweet spoiled child. But in his eyes I'm horrible that I ask his son to learn how to grow up to be a responsible, well manner, well groom (or at least hygienic in some way). The worse part is he has a mother!!! I don't speak with the mother because, as your situation, this child is SPOILED AND ENABLED by everyone. Mom, dad, grandparents and etc. But in our home how dare he have to do anything. Even school work, I barely ever saw him do any school work and I was the only one who bring up "does SS have homework, or does SS have e-learning to do?" I get the "oh yeah that's right!" Response from dad. Like if I never mentioned this he wouldn't do it. I am so beyond how this kid is allowed to move onto the next grade because he hasn't done anything. Just like your situation, again, this kid has no decent sleep time or  even wakes up at a normal hour. During "actual" school nights he would go to bed at 10:30/11pm, just to wake up at 6:30am. Now that the pandemic has started this kid thinks it's a vacation. He is on a screen alll day and all night. If he's not on his PS4 he has his phone. I mentioned this to dad and he agrees but NEVER executes anything!!! He ignores it and allows his spawn to do whatever he wants.

I eventually got fed up being the only one who felt like he has no structure, no responsibility, takes no accountability for any bad behavior or for mistakes, does not know his left to his right hand!!  I disengaged completely- totally resenting this child because of his enabling father!!! It even got to the point that I kicked them out because I couldn't take having a child like this in my home where he got more authority over me and also has no rules, yet my daughter had rules and chores and etc. All the while his son had absolutely nothing and felt like he was to have special treatment. That's not what I signed up for. 
 

Like you in the beginning I was so in love with his son, I did everything. I was a maid, a cook, a chauffeur, a teacher, a laundry mat and anything you can think of for being a second mom to him (although I'm not his mother) But I was the bad guy who asked for such simple things that are so beyond normal for a child his age DAD though I was mean to ask such things and never seen it as a positive. I felt disrespected, ignored and as if I have NO authority in my own house! I told him to leave and take his son with: I said I don't care if his son wants to stink like crap, with dirty hands smell breath, if he wants his son to be a slob, if he feel raising a child on electronics is his way of upbringing a young man into this world. This is his problem, his son. He can feed him junk and continue to allow his picky eating habits not my issue anymore! He can be his friend all he wants not a father a friend. I expressed to his father that he is a weak parent and he's afraid of setting rule because he will think his son will hate him, which he won't, but I let it all out. At that point I was ready for a divorce- I didn't care! I am not raising this child I have my own to raise and she is nowhere near like this boy! 
 

Since him moving out, it was months before I allowed him and his spawn child back! I didn't force him to change or take my side or anything. I simply told him that I am not in a position to fee like I'm should be disrespected and or have zero authority in my home, especially when it comes to a child!!  He wants to be a parent go parent and have his son act like that elsewhere, see how everyone else feels about all this. 
 

Low and behold my husband ended up staying with his mother. Even his mother over time noticed how spoiled and enabled her grandchild was and how dear ole dad coddled him from even grandma for her trying to set rules too. Because everyone in her home works! She told my husband that his son needs a decent bedtime because she doesn't want to hear his son on his games or even hear his phone at night! She even told my husband that he should learn to pick up after himself because she is tired of picking up her grandsons mess. I got a call from his mother (thinking she hated me cause I kicked him out) but she asked what was going on? How do I feel? And or what happened. I seriously let out everything to her. It's not like I hate his son. I actually really love him, and as my own, I care about this child. But I refuse to be with his father if he feels this is what he think it is like to raise a child. I don't want children with him either because he will be the same man and father/friend! I told her how I tried to set some rules and boundaries, at least a small balance of structure! I was the evil one! I expressed it all. Her response was the same. She sees how her son is raising a spoiled enabled child! She is getting fed up to having to care for her grandson because he does and knows nothing. NO ONE EVER TELLS HIM "NO"! She wanted to see if she could help because she wants my husband out and her grandson to go back with his mother. Although she loves them both, her child (my husband) is older and should be raising a young man not a spoiled rotten child and she can't sit and watch what's going on (she too felt like she was losing authority in her home). The problem was her son (my husband).

So my advice to you is after reading my long story. Is that you are the bad guy and it is your BF who is using you as a babysitter. He not looking for a mom to raise his son. He's looking for someone to blame so he can be to good guy in raising his kid! He doesn't want to know or hear that he is the problem and his son will grow up to be a degenerate because he was spoiled and enabled all his life! As where you, the amazing woman who lived and cared for both of them was only trying to help raise a fine young man. A be a great GF. It isn't your responsibility to care for his son, it is his son, it just seemed so convenient since you are there that he expects you to take on such a huge responsibility to care for his son while he works and or etc. This child is 10, he's not 2 or 4 years old. He's 10! He needs to be consistent, on him to learn how to do simple things as a young boy! But you seem to be the only one doing this while his father is lacking how to parent his own son! I will tell you now, that if you stay you WILL resent this child and you will be upset and frustrated at his father for being this Disney dad! You will lose any authority in the home you are both in. At some point you will want to leave! It won't get better. So one or two things : stay and do what you can to discuss with dad that his parenting skills suck (not in those words) but explain how you feel. Ask for his help since this is his son. Or leave. 
 

sorry this was so long. But the problem isn't you. It's the father! His son will just continue to be the way he is if his father allows it and makes you like you are the bad guy. He will always be trophy dad 

I feel like you are falling into a trap. Because you are financially dependent, you lack power in the relationship. You are in the role of doing all the work of parenting, and being rewarded by being the "bad guy." He gets to swoop in and be the fun dad by undermining your efforts. All the while, you feel like you are being observed and tested, based on your "parenting." It's a lose-lose situation for you. If you don't set appropriate limits, you get to live with a feral child and spend your time as a servant. If you do set limits, you are seen as not being caring and nurturing enough. 

To me, rules should be agreed upon by both of you. Have *him* be the one who communicates them to the kids, preferably with you right there. He has to enforce them as much as you do. I would argue that he should be the primary enforcer since the child is already 10, with your efforts backing up what he has already set in place. If he is unwilling to do this, you cannot thrive in this situation and should leave. 

I sort of know how you feel. Idk if men (or just some of them) gravitate toward this role of provider and fun parent. My SO does. From listening to his stories , his last gf was put in the position you are in. It ended badly. He said she was too strict but he did not "see the love" from her. I think it was due to unfair expectations and ideas about SMs. If non-bioparents are strict it's seen as not liking the child or picking on the child. I will not be placed in that position and neither should you. 
 

He does not want to parent his son.  He rather be friends with his DS.  He does not want to inforce any house rules.  

As bed time, screen time.  SS understand if he gives DH a hard time, screaming, DH will cave in.  You who are normal, are caught in the middle.  When SS starts to fail, it's going to be your fault.  
Most of us love our SK, but not the type of love that   SK are a reflection of me type of love. What only a bio parent can have, but bio parents think all SP should have.  

Your real problem is you can't disengage, because then SS will be running the house.  Stop cooking special meals, if DH wants SS to have special meals he can make it.  Then actually he can do all the dinner. Inforce rules. If he doesn't go to bed on time the next night it's 1/2 hours earlier. Ect  
 

Rescue projects never work out for the rescuer.  Both this kid and his father are rescue projects.   You are attempting to rescue this kid from the failed parenting he has suffered at the hands of his father and deceased mother.  Since his father remains alive.... you can't rescue this kid from the crappy parenting daddy subjects him to.

You can't rescue this idiot father from himself.

So..... give some serious long term thought to the impact to your life of a future of having to deal with both an idiot failed man and father and the results of his failure as a parent.

You are struggling with these issues with the 10yo because of 10 years of crappy parenting by your SO and the BM.

Stop filling in for their crappy parenting and let the surviving parent deal with trying to stuff food down the gullet of a picky eater of a 10yo, stuffing school material in this kid's ear, purging screens from the life of this game and television addicted kid and getting him onto a sane and healthy sleep schedule.  

This idiot has been clear that he considers you and the rules you are attempting to invoke in the life of his 10yo to be the problem. Not only that, he  gaslights you on that topic rather than having the maturity and balls to actually engage and speak with you on the topic. He is not nearly the adult that can participate in resolving these issues for his son.  And.. it really is not your problem unless you naively continue to sacrifice yourself on the alter of SParental martyrdom to the deceased BM, the idiot father and their tragically doomed spawn.

Value yourself and live your life.  There are partners out there who are not tragically flawed that you can make a life with.  Some with prior relationship kids and some without.  Find one.

Inform daddy that no longer will you either get up with his kid, educate his kid, feed his kid, or deal with putting his kid to bed. Since Dadidiot thinks you are the problem, give him an immediate and permanent lesson that he is the problem and the his son is the result of his failed parenting and that of his dead wife.

Pack up and get on with your life.

At the very least review your original post and think of what advice you would give to someone asking you for your advice in this situation.  Change is almost impossible for most people.  It is too late for this man and it is so close to too late for this 10yo that the odds of success are slim and none.

Good luck.  Take care of you.